Critical temperature in the two-layered Ising model

Using the transfer-matrix mean-field approximation we precisely calculate the critical temperature in the two-layered Ising model. When the intra-layer interactions are the same, our estimations of the shift exponent φ are in agreement with some scaling arguments which predict φ=γ, where γ is the susceptibility exponent. However, for unequal intra-layer interactions our result φ=0.5 show that scaling arguments prediction φ=γ/2 is incorrect. In this case an approximate decimation scheme is proposed which gives φ consistent with numerical calculations.